# Prove: If $\sum a_n,\sum b_n$ are convergent series of nonnegative terms, then $\sum\sqrt{a_nb_n}$ converges.

I am trying to use the fact that $\lim_{n \rightarrow \infty} \sqrt{s_n} = \sqrt{s}$ where $s_n$ is the sequence of partial sums of $\sum a_n$, the same for $\sum b_n$, then I apply algebra of limits, and that is all, but I am not sure about my process.

• Hi and welcome to Math.SE. Please update your question using mathjax. It will make it easier for people to read. – mickep Sep 14 '15 at 18:29
• Are you assuming that $a_i, b_i$ are each nonnegative? Otherwise $\sqrt{a_nb_n}$ is a problem. – vadim123 Sep 14 '15 at 18:30
• yes, ai,bi are nonnegative. – Iván Galeana Aguilar Sep 14 '15 at 18:33
• Hint: $\sqrt{a_nb_n}\leq \max(a_n,b_n) \leq a_n+b_n$ – Milo Brandt Sep 14 '15 at 18:35
• I understand you define $s_n=\sum_{j=1}^n a_j$. Then you write "the same for $b_n$"; do you here mean that you would like to set $s_n=\sum_{j=1}^n b_j$ too? what do you have in mind exactly? :-) – Math-fun Sep 14 '15 at 18:56

Hint: Use that $$\sqrt {a_n b_n} \leq \frac{a_n + b_n}{2}$$
if $a_n, b_n$ are non-negative. Use the Comparison Test.